Finite Time Reachability Partial Control

• Actuator Failures: Some control inputs become unavailable
• Degraded Performance: Remaining actuators operate at reduced capacity
• Underactuated Dynamics: System has fewer controls than degrees of freedom
• Safety-Critical: Must reach target despite failures

Time Horizon Partitioning

1. Successive Intervals: Divide total time into decreasing-length segments
2. Approximate Dynamics: Build linear driftless model at each segment start
3. Control Design: Design inputs based on approximate dynamics
4. Recursive Application: Apply to each interval until target reached

Linear Driftless Approximation

• Simplified model: ẋ = B(x₀)u (no drift term)
• Valid locally around initial state x₀
• Enables analytical control design
• Error bounded by interval length

Methodology

Algorithm Overview

Input: Initial state x₀, Target state x*, Total time T
Output: Control input u(t) for t ∈ [0,T]

1. Partition [0,T] into intervals [t₀,t₁], [t₁,t₂], ..., [tₙ₋₁,tₙ]
   with decreasing lengths
2. For each interval [tᵢ, tᵢ₊₁]:
   a. Build linear driftless approximation at x(tᵢ)
   b. Design control to drive toward x*
   c. Apply control and observe resulting state
3. Return to target if not reached

Practical Applications

Use Cases

1. Aircraft Control: Emergency landing with actuator failures
2. Spacecraft Maneuvering: Reach target orbit with thruster failures
3. Robotic Manipulation: Complete task with joint actuator failure
4. Vehicle Control: Navigate to safe stop with brake/steering failure

Safety-Critical Systems

• Guaranteed reachability despite failures
• Explicit computation of reachable set
• Constraint satisfaction (state and input bounds)
• Finite-time guarantees (not asymptotic)

Theoretical Guarantees

Key Results

• Reachability: System can reach target if it lies in reachable set
• Constraint Satisfaction: State and input constraints respected
• Finite Time: Explicit bound on time to reach target
• Partial Control: Works with reduced control authority

Limitations

• Requires knowledge of system dynamics (even if nonlinear)
• Computational cost for high-dimensional systems
• Conservative reachable set estimates
• May fail if failure is too severe

Activation Keywords

• finite-time control
• reachability
• partial control
• constrained systems
• nonlinear control
• actuator failure
• emergency control
• driftless approximation

References

• Padmanabhan, R., & Ornik, M. (2026). Finite-time Reachability for Constrained, Partially Uncontrolled Nonlinear Systems. arXiv:2604.08327.

Tools Used

• exec
• read
• write

Instructions for Agents

1. 理解需求：分析用户请求的具体场景
2. 选择方法：根据上下文选择合适的技术方案
3. 执行操作：按照技能描述实施具体步骤
4. 验证结果：检查结果是否符合预期

Examples

Example 1: Basic Usage

User: 请帮我应用此技能

Agent: 我将按照标准流程执行...

Example 2: Advanced Usage

User: 有更复杂的场景需要处理

Agent: 针对复杂场景，我将采用以下策略...